From 1-homogeneous Supremal Functionals to Difference Quotients: Relaxation and Γ-convergence

In this paper we consider positively 1-homogeneous supremal functionals of the type F (u) := supΩ f(x,∇u(x)). We prove that the relaxation F̄ is a difference quotient, that is F̄ (u) = RF (u) := sup x,y∈Ω, x6=y u(x)− u(y) dF (x, y) for every u ∈ W (Ω), where dF is a geodesic distance associated to F . Moreover we prove that the closure of the class of 1-homogeneous supremal functionals with respect to Γ-convergence is given exactly by the class of difference quotients associated to geodesic distances. This class strictly contains supremal functionals, as the class of geodesic distances strictly contains intrinsic distances.